The Loop Atlas: A loop comes home to where it started. What did it leave behind?

The Loop Atlas follows one question across a whole landscape of machines and bodies: when a loop comes home to where it started, what did it leave behind? Each loop is told in plain words with a toy behind every claim, the knobs a builder would turn, every prediction registered before its run with its verdict kept, and every miss kept with a redesign. The book comes with a companion volume drawn from the same register of results, The Loop Atlas: the record, which prints every prediction a run judged, every other check a run made, how each came out, and every miss with its reason. The files. Open the reader's edition first (LOOP_ATLAS_READERS_EDITION_2026-10-19.pdf, with its source as md). Beside it are the eighteen toy pages and the front page that maps them, each an HTML file that opens in a web browser; and the archive (LOOP_ATLAS_ARCHIVE_2026-10-31.zip), every tool, run and figure, with a short map, README.md, at its top. Among the pages, start with the front page, LOOP_ATLAS_EXPLORER_FRONT_2026-11-09.html: it draws the book as a tree and hangs every page on the chapter that tells it. The companion volume, The Loop Atlas: the record (LOOP_ATLAS_RECORD_2026-10-29.pdf, with its source as md), is deposited as a record of its own, named among this record's related works. The toys. Eighteen interactive pages travel with the book, each a single HTML file that opens offline in any browser: A Skater Who Never Pushes Off (LOOP_ATLAS_SKATER.html): a skater who never lifts a foot swizzles forward; two knobs, toe swing and leg push, make an engine when they move in step, and nothing when they move a quarter cycle apart. Running Through the Air (LOOP_ATLAS_EXPLORER_ROOM1_2026-10-17.html): a long jumper's arm circles and hitch-kick, and how much forward tip each one takes back before landing. The Diver's Clock (LOOP_ATLAS_EXPLORER_ROOM2_2026-10-17.html): a platform dive laid out as a clock of moves; kick out by the count or by the cue, and see which lands cleaner. Holonomy the Cat (LOOP_ATLAS_EXPLORER_ROOM3_2026-10-17.html): a cat that slips off a branch upside down with no spin and turns over on the way down, with a twisting cat and a twist-free cat side by side. The Stretch and Squeeze Explorer (LOOP_ATLAS_STRETCH_SQUEEZE_EXPLORER_2026-10-13.html): a needle with memory, driven by a pulse, settles onto one orbit; start a small circle of nearby needles anywhere on it, carry it round, and watch it pulled out long and thin, then crushed, red where it stretches and blue where it squeezes; adapted from the Depth Axis Explorer of A Rider in Two Waves. The Torus of Fractions (LOOP_ATLAS_TORUS_OF_FRACTIONS_2026-10-30.html): every lock the needle finds, p turns every q beats, drawn as one closed loop on a doughnut; set out as a forest, a doughnut for each loop in rows by generation, or all on one shared doughnut, or nested; the odd over odd, even over odd and odd over even switched on and off; any fraction's two parents and its children lit with a click; seven checks run as it opens. The Two-Gear Hill (LOOP_ATLAS_EXPLORER_ROOM4_2026-10-31.html): stand a truck on a hill with a wave passing, guess which lane it ends in, let it go, and race it against another. Latches and Switches (LOOP_ATLAS_EXPLORER_ROOM5_2026-10-31.html): turn one knob of a bead's ear at a time, and race the beads along the soup. The Long Snake (LOOP_ATLAS_EXPLORER_ROOM6_2026-10-31.html): race two snakes on the same swell, by equal time or equal fuel, each on its own knobs. Rings That Leapfrog (LOOP_ATLAS_EXPLORER_ROOM7_2026-11-02.html): fly smoke rings as the runs flew them beside one ring alone on the same fuel, and race one train of rings against another. Tied Bells in a Remembering Water (LOOP_ATLAS_EXPLORER_ROOM8_2026-11-03.html): find the waters where the rings pay two tied bells, and swim them beside plain water and a rival water. Heat in a Small Piece (LOOP_ATLAS_EXPLORER_ROOM9_2026-11-03.html): set a warm brick between two blocks of ice, guess whether its middle goes colder than the ice, and race a brick of another length. The Needle's Stretch and Squeeze (LOOP_ATLAS_EXPLORER_ROOM10_2026-11-04.html): carry a hair's difference round the pulsed needle's lap, guess how big it grows, and race another start. The Machine on the Grid (LOOP_ATLAS_EXPLORER_ROOM11_2026-11-08.html): set a generator's load and its inertia, knock it out of step, guess whether it comes back by itself, and race it against another; the swing equation runs in the page, drawn as a ball on a tilted washboard. The Wave Soup and Its Ears (LOOP_ATLAS_EXPLORER_ROOM12_2026-11-08.html): choose how a bead's ear looks and what it reads between looks, guess whether it keeps its headway, race one ear against another, and watch 64 riders surf a crest with waves coming the other way. The Maintenance Meter (LOOP_ATLAS_EXPLORER_ROOM13_2026-11-08.html): pick a fleet of machines and a maintenance reminder, guess whether it throws away less of each life than the calendar, race it against another, see where a wearing needle's cliff sits in each room, and set a slowing rider's two clocks side by side. The Column and the Hiker (LOOP_ATLAS_EXPLORER_ROOM14_2026-11-08.html): set a hiker's pack and how often he looks up, follow a column of walkers a metre apart at night by sight or by touch, read the cat's eyes on the pack ahead late, and pass a halt back man to man or order it at once, guessing first and racing one halt against another. The First Swimmer and Its Faces (LOOP_ATLAS_EXPLORER_ROOM15_2026-11-08.html): make a three-piece swimmer's head, middle or tail broad, see how far each body goes a lap and for the work the water takes, guess whether it goes farther for the work than the round swimmer, race one body against another, and broaden the end pieces or the middle from grip 1 to 10. Holonomy the Cat is the book's cover. The jumper, the diver and the cat each offer four ways in: a way to play (the jumper's is to build a jump with a coach's knobs, the diver's to lay out a dive, the cat's to launch it from its tree), a way to look, a way to learn by guessing first, and a check, which reruns the book's numbers in the browser and prints each beside the value of record. The twelve later rooms check themselves the same way as they open, each against its own runs. What it found. 1,104 registered predictions judged, 706 held; 51 meetings with the published record, from Feigenbaum's constant to Kane and Scher's falling cat and Tong and Dullin's diver, rebuilt within 0.15 per cent once one entry their published table most likely drops is restored; and results not yet seen in print, the first of them the mirror between a twisting and a twist-free falling cat, claimed as new on a limited, noncommercial review of the citations and references, and, more modestly, a listening swing rider whose mistuning costs exactly a mismatched line's share, a registered law for light carried round a loop of filters and a chaos that keeps its gear better than a lock near its edge, none found in print in the pages reached; machines timed by their own events rather than a clock, a listening swing rider that works on swings from 0.5 to 12.5 metres long, an ear that harvests an even soup of waves that a clock-timed switch cannot, a drift of 0.124 against 0.009, and a spring ear that stops hearing exactly where its own ring dies, at critical damping; smoke rings that leapfrog, 14 to 19 per cent faster than one ring on the same fuel; a fixed maintenance alarm read where nothing hides the wear, wasting 0.031 to 0.051 of a machine's life where the calendar wastes 0.250 to 0.333; a diver who kicks out on the body's angle, missing 0.420 as much as one who counts, a number predicted before the run; a latch that holds until the push turns, earning up to 1.82 times the push of a trigger without memory; and for every machine the knobs a builder would turn, in order, with what each trades. Quadrature, two shape coordinates a quarter cycle apart, was the method that guided the experiments and the reading of their results. Limits. A preprint, not peer reviewed, with a limited literature pass. Toy models with assumed numbers, not experiments. Built by an independent researcher with an AI, in the open. Contact. Matthew "Bear" Schulz, Independent Researcher, Belmont, CA · ORCID 0009-0000-0956-5743 · mattschulz97@protonmail.com